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Hexi Ye

Publications and source records attributed to Hexi Ye.

14 recordsLinked to original sources
arXiv2024

Bounded geometry for PCF-special subvarieties

For each integer $d\geq 2$, let $M_d$ denote the moduli space of maps $f: \mathbb{P}^1\to \mathbb{P}^1$ of degree $d$. We study the geometric configurations of subsets of postcritically finite (or PCF) maps in $M_d$. A complex-algebraic subvariety $Y \subset M_d$ is said to be PCF-special if it contains a Zariski-dense set of PCF maps. Here we prove that there are only finitely many positive-dimensional irreducible PCF-special subvarieties in $M_d$ with degree $\leq D$. In addition, there exist constants $N = N(D,d)$ and $B = B(D,d)$ so that for any complex algebraic subvariety $X \subset M_d$ of degree $\leq D$, the Zariski closure $\overline{X\cap\mathrm{PCF}}~$ has at most $N$ irreducible components, each with degree $\leq B$. We also prove generalizations of these results for points with small critical height in $M_d(\bar{\mathbb{Q}})$.

math.DS↗
arXiv2021

Common preperiodic points for quadratic polynomials

Let $f_c(z) = z^2+c$ for $c \in \mathbb{C}$. We show there exists a uniform bound on the number of points in $\mathbb{P}^1(\mathbb{C})$ that can be preperiodic for both $f_{c_1}$ and $f_{c_2}$ with $c_1\not= c_2$ in $\mathbb{C}$. The proof combines arithmetic ingredients with complex-analytic; we estimate an adelic energy pairing when the parameters lie in $\bar{\mathbb{Q}}$, building on the quantitative arithmetic equidistribution theorem of Favre and Rivera-Letelier, and we use distortion theorems in complex analysis to control the size of the intersection of distinct Julia sets. The proof is effective, and we provide explicit constants for each of the results.

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arXiv2019

Uniform Manin-Mumford for a family of genus 2 curves

We introduce a general strategy for proving quantitative and uniform bounds on the number of common points of height zero for a pair of inequivalent height functions on $\mathbb{P}^1(\overline{\mathbb{Q}}).$ We apply this strategy to prove a conjecture of Bogomolov, Fu, and Tschinkel asserting uniform bounds on the number of common torsion points of elliptic curves in the case of two Legendre curves over $\mathbb{C}$. As a consequence, we obtain two uniform bounds for a two-dimensional family of genus 2 curves: a uniform Manin-Mumford bound for the family over $\mathbb{C}$, and a uniform Bogomolov bound for the family over $\overline{\mathbb{Q}}.$

math.NT↗
arXiv2017

Quasi-adelic measures and equidistribution on $\mathbb{P}^1$

Baker-Rumely and Favre-Rivera-Letelier independently proved an important arithmetic equidistribution theorem for points of small height on the Berkovich compactification of the projective line with respect to an adelic measure on $\mathbb{P}^1$. Around the same time, Chambert-Loir proved a more general version of this arithmetic equidistribution theorem in the setting of curves from a different approach. We generalize the notion of an adelic measure to that of a quasi-adelic measure on $\mathbb{P}^1$, and show that arithmetic equidistribution of points with small height holds for quasi-adelic measures as well. Moreover, we show that the canonical measure associated with a dynamical pair $(f,c)$ on $\mathbb{P}^1$ is rarely adelic. We prove that for certain examples of families of rational functions parameterized by $\mathbb{P}^1$, corresponding to the curve $\mathrm{Per}_1(λ)$ introduced by Milnor for a root of unity $λ$, the measure corresponding to a general starting point is quasi-adelic. Finally, we place our results in context by establishing their connection with two problems in arithmetic dynamics.

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arXiv2017

Bounded height in families of dynamical systems

Let a and b be algebraic numbers such that exactly one of a and b is an algebraic integer, and let f_t(z):=z^2+t be a family of polynomials parametrized by t. We prove that the set of all algebraic numbers t for which there exist positive integers m and n such that f_t^m(a)=f_t^n(b) has bounded Weil height. This is a special case of a more general result supporting a new bounded height conjecture in dynamics. Our results fit into the general setting of the principle of unlikely intersections in arithmetic dynamics.

math.NT↗
arXiv2016

The Dynamical Manin-Mumford Conjecture and the Dynamical Bogomolov Conjecture for split rational maps

We prove the Dynamical Bogomolov Conjecture for endomorphisms of P^1\times P^1 defined over a number field. We use the equidistribution theorem for points of small height with respect to an algebraic dynamical system, combined with a theorem of Levin regarding symmetries of the Julia set. Using a specialization theorem of Yuan and Zhang, we prove the Dynamical Manin-Mumford Conjecture for endomorhisms of P^1\times P^1 defined over the complex numbers.

math.NT↗
arXiv2016

The Dynamical Andre-Oort Conjecture for cubic polynomials

In the moduli space of degree d polynomials, the special subvarieties are those cut out by critical orbit relations, and then the special points are the post-critically finite polynomials. It was conjectured that in the moduli space of degree d polynomials, the subvarieties containing a Zariski-dense set of special points are exactly these special subvarieties. In this article, we prove the first non-trivial case for this conjecture: the case of cubic polynomials.

math.NT↗
arXiv2015

Bifurcation measures and quadratic rational maps

We study critical orbits and bifurcations within the moduli space of quadratic rational maps on $\mathbb{P}^1$. We focus on the family of curves, $Per_1(λ)$ for $λ$ in $\mathbb{C}$, defined by the condition that each $f\in Per_1(λ)$ has a fixed point of multiplier $λ$. We prove that the curve $Per_1(λ)$ contains infinitely many postcritically-finite maps if and only if $λ= 0$; addressing a special case of [BD2, Conjecture 1.4]. We also show that the two critical points of a map $f$ define distinct bifurcation measures along $Per_1(λ)$.

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arXiv2015

The Dynamical Andre-Oort Conjecture: Unicritical Polynomials

We establish the equidistribution with respect to the bifurcation measure of post-critically finite maps in any one-dimensional algebraic family of unicritical polynomials. Using this equidistribution result, together with a combinatorial analysis of certain algebraic correspondences on the complement of the Mandelbrot set $M_2$ (or generalized Mandelbrot set $M_d$ for degree $d>2$), we classify all complex plane curves $C$ with Zariski-dense subsets of points $(a,b)\in C$, such that both $z^d+a$ and $z^d+b$ are simultaneously post-critically finite for a fixed degree $d\geq 2$. Our result is analogous to the famous result of Andre regarding plane curves which contain infinitely many points with both coordinates CM parameters in the moduli space of elliptic curves, and is the first complete case of the dynamical Andre-Oort phenomenon studied by Baker and DeMarco.

math.AG↗
arXiv2014

Torsion points and the Lattes family

We give a dynamical proof of a result of Masser and Zannier [MZ2, MZ3] about torsion points on the Legendre family of elliptic curves. Our methods also treat points of small height. A key ingredient is the arithmetic equidistribution theorem on $\mathbb{P}^1$ of Baker-Rumely, Chambert-Loir, and Favre-Rivera-Letelier. Torsion points on the elliptic curve coincide with preperiodic points for the degree-4 Lattes family of rational functions. Our main new results concern properties of the bifurcation measures for this Lattes family associated to marked points.

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arXiv2014

Rational functions with identical measure of maximal entropy

We discuss when two rational functions $f$ and $g$ can have the same measure of maximal entropy. The polynomial case was completed by (Beardon, Levin, Baker-Eremenko,Schmidt-Steinmetz, etc., 1980s-90s), and we address the rational case following Levin-Przytycki (1997). We show: $μ_f = μ_g$ implies that $f$ and $g$ share an iterate ($f^n = g^m$ for some $n$ and $m$) for general $f$ with degree $d \geq 3$. And for generic $f\in \Rat_{d\geq 3}$, $μ_f = μ_g$ implies $g=f^n$ for some $n \geq 1$. For generic $f\in \Rat_2$, $μ_f = μ_g$ implies that $g= f^n$ or $σ_f\circ f^n$ for some $n\geq 1$, where $σ_f\in PSL_2(\C)$ permutes two points in each fiber of $f$. Finally, we construct examples of $f$ and $g$ with $μ_f = μ_g$ such that $f^n \neq σ\circ g^m$ for any $σ\in PSL_2(\C)$ and $m,n\geq 1$.

math.DS↗
arXiv2012

Finiteness of commutable maps of bounded degree

In this paper, we study the relation between two dynamical systems (V,f) and (V,g) with f. g = g . f. As an application, we show that an endomorphism (respectively a polynomial map with Zariski dense, of bounded Pre(f) has only finitely many endomorphisms (respectively polynomial maps) of bounded degree which are commutable with f.

math.NT↗
arXiv2011

The Schwarzian derivative and polynomial iteration

We consider the Schwarzian derivative $S_f$ of a complex polynomial $f$ and its iterates. We show that the sequence $S_{f^n}/d^{2n}$ converges to $-2(\partial G_f)^2$, for $G_f$ the escape-rate function of $f$. As a quadratic differential, the Schwarzian derivative $S_{f^n}$ determines a conformal metric on the plane. We study the ultralimit of these metric spaces.

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